Data driven regularization by projection

We demonstrate that regularisation by projection and variational regularisation can be formulated in a purely data driven setting when the forward operator is given only through training data. We study convergence and stability of the regularised solutions. Our results also demonstrate that the role of the amount of training data is twofold. In regularisation by projection, the amount of training data plays the role of a regularisation parameter and needs to be chosen depending on the amount of noise in the measurements. In this case using more data than allowed by the measurement noise can decrease the reconstruction quality. In variational regularisation, however, the amount of training data controls the approximation error of the forward operator and hence more training data always results in better reconstructions.